In Figure REF (a), we show an example point cloud, consisting of discrete points in the shape of an overturned dumbell. When \(\alpha \)  is large (e.g., \(\alpha \approx 50\)  in Figure REF ), the boundary of the \(\alpha \) -shape is nearly the convex hull of our data, and we miss meaningful structure in the dumbell shape. Indeed, when \(\alpha = \infty \) , the \(\alpha \) -shape is simply the convex hull of the point cloud. As we decrease \(\alpha \)  in Figure REF (b), our \(\alpha \) -shapes look more like hourglasses and eventually pick up the shape of the dumbell between \(\alpha = 10\)  and \(\alpha = 5\) . When \(\alpha = 0\) , the \(\alpha \) -shape is just the data itself [1]}. (Notably, \(\alpha \) -shapes are also defined for \(\alpha < 0\) ; see [2]}, [1]} for more information.) While we do not discuss it further here, there is a connection between \(\alpha \) -shapes, Voronoi diagrams, and Delaunay triangulations [4]}, [2]}, [6]}. Additionally, \(\alpha \) -shapes are related to \(\alpha \) -complexes, which—like the Vietoris–Rips complex that we introduced in §REF —can serve as the basis for computing persistent homology [7]}, [6]}, [1]}. In §REF , we apply \(\alpha \) -shapes to extract stripe–interstripe boundaries and quantify pattern curviness.
